In developing an investment portfolio, a number of assets are selected from a large group of available assets and the amount of capital to be invested is allocated among the selected assets. An objective of portfolio selection and allocation is to maximize the return on the investment with a controlled degree of risk. It is generally understood that by diversification, the expected degree of risk associated with a portfolio can be controlled.
There are several well known mathematical modeling techniques for portfolio selection and management, such as those which employ the Markowitz portfolio selection model. The Markowitz model is a mean-variance model which shows that for a given upper bound on risk that an investor is willing to accept or a lower bound on the acceptable level of return, optimal portfolio selection can be performed by solving a quadratic optimization problem. When it is assumed that risk and return are exactly known and noise free, this relationship can be expressed graphically to define what is known as the “efficient frontier” as illustrated in FIG. 1 by line 105. An extension of the Markowitz mean-variance model is the capital asset pricing model (CAPM) developed by Sharpe et al. The import of the theoretical works of Markowitz and Sharpe were recognized in 1990, as these individuals shared the Nobel prize in Economic Science for their work on portfolio allocation and asset pricing.
Although the Markowitz mean-variance model is important theoretically, as a practical matter, many practitioners have shied away from this technique due to inherent “real world” uncertainties. For example, the solutions to the optimization problems are often sensitive to perturbations in the underlying market parameters of the problem, and since these market parameters are estimated from noisy data which are subject to statistical variations, the resulting optimizations may not be reliable. As a result, in practice, the so-called “efficient frontier” is not an error free line defining the relationship between risk and return which is exactly known, but is an unknown, error-bounded region, such as is illustrated by the shaded region 110 in FIG. 1. As set forth in U.S. Pat. No. 6,003,018 to Michaud et al., simulations can be used to estimate the error in the frontier region. However, such an approach suffers from the “curse of dimensionality” and will be rendered ineffective as the number of assets increases.
Accordingly, there remains a need for systems and methods which can efficiently provide robust portfolio allocations within a defined confidence interval.